How many bijections \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) are there such that \(f(x+y)=f(x)+f(y)\) for…
How many bijections \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) are there such that \(f(x+y)=f(x)+f(y)\) for all \(x, y \in \mathbf{Z}\) ?
One
Two
Three
Infinitely many
Solution
\(f: \mathrm{Z} \rightarrow \mathbf{Z}\)
\(\begin{aligned}
& f(x+y)=f(x)+f(y) ; x, y \in \mathbf{Z} \\
& \therefore \quad f(x)=k x \\
\end{aligned}\)
So, there are infinitely many bijections